Introduction
Abaqus supports many different creep models, for example:
- Time Law
- Time Power Law
- Strain Law
- Power Law
- Hyperb Law
- Double Law
- Darveau Law
All of these creep models can be used with a linear elastic, or a linear elastic with plasticity model. In this article I will discuss some of the details of these creep laws, and show some of the limitations of this model framework.
Time and Time Power Creep Models
➡️ The classical Time creep model in Abaqus has the following equation:
\(\dot{\varepsilon}_c = A q^n t^m\),
where \(\dot{\varepsilon}_c\) is the creep rate magnitude, \(q\) is the effective stress driving the creep (which by default is the Mises stress), and \(t\) is the time (since the start of the simulation or the start of the creep step). The parameters \([A, n, m]\) are material parameters.
➡️ There is also a newer version of this model available called the Time Power model. This model has the following creep equation:
\(\dot{\varepsilon}_c = \displaystyle \dot{\varepsilon}_0 \left(\frac{q}{q_0}\right)^n \left(\dot{\varepsilon}_0 t \right)^m\).
In this equation \([\dot{\varepsilon}_0, q_0, n, m]\) are material parameters.
The problem with the original Time creep model is that the parameter A has odd units and can have a value that is super small. For example, if you are using SI units and the applied stress is 100 MPa, and \(n=5\), then \(q^n=10^{40}\). If the creep rate is slow at that stress then the parameters \(A \approx 10^{-40}\), which can cause floating point precision problems. In other words, don’t use the Time creep model. It is included in Abaqus only due to historical reasons.
Strain and Power Creep Models
The classical Strain creep model has the following equation:
\(\dot{\varepsilon}_c = \left[A^{1/(m+1)} (m+1)^{m/(m+1)}\right]\, q^{n/(m+1)}\, \varepsilon_c^{m/(m+1)}.\)
Note that I wrote the equation a bit differently than what is used in the Abaqus Manuals in order to show how the equation is separated into three terms: a pre-factor, the stress raised to a constant, and the creep strain magnitude raised to a constant. Just like the Time creep model, this model can also suffer from rounding point precision problems for certain unit systems. For that reason, Abaqus implemented the Power creep model with the following equation:
\(\dot{\varepsilon}_c = \displaystyle \dot{\varepsilon}_0 \left(\frac{q}{q_0}\right)^{n/(m+1)} \left( (m+1)\varepsilon_c \right)^{m/(m+1)}.\)
This model is properly normalized by \(q_0\), and should be used instead of the classical Strain model.
Hyperb, Double, and Darveau Creep Models
Abaqus also supports a few additional temperature-dependent creep models that were specifically developed for high-temperature creep of metals. The Hyperb creep model has the following creep equation:
\(\dot{\varepsilon}_c = \displaystyle A \left( \sinh(Bq) \right)^n \exp\left( \frac{-\Delta H}{R(\theta – \theta_Z)}\right).\)
In this equation \(sinh(x)=(\exp(x)-\exp(-x))/2\) is the hyperbolic sine function, \(R\) the gas constant, \(\theta\) the temperature, and \(\theta_Z\) the absolute zero temperature. I find it odd that they did not use \(q_0 = 1/B\) instead. The Double creep model has the following equation:
\(\dot{\varepsilon}_c = \displaystyle A_1 \exp\left( \frac{-B_1}{\theta-\theta_Z} \right)\left[\frac{q}{\sigma_0} \right]^{C_1} + A_2 \exp\left( \frac{-B_2}{\theta-\theta_Z} \right)\left[\frac{q}{\sigma_0} \right]^{C_2}.\)
This model has two creep terms to allow more flexibility when fitting the creep response. Finally, the Darveau model has the following creep equation:
\(\dot{\varepsilon}_c = \displaystyle \dot{\varepsilon}_s \left[ 1 + \varepsilon_TB\exp(-B \dot{\varepsilon_s} t) \right],\)
where
\(\dot{\varepsilon}_s = \displaystyle C_{ss}[ \sinh(\alpha q)]^n \exp\left( \frac{-Q}{R(\theta-\theta_Z)} \right),\)
is the steady state creep rate.
Stress-Strain Predictions
After showing all those equations, let’s take a look at the type of predictions you can get using a linear elastic material model combined with a creep model. The following figure shows that the predicted creep strain is increasing linearly with time for the Power creep model if the m-parameter is 0. This is exactly what is expected based on the creep equation.
If I change the m-parameter to -0.4 then the predicted creep strain is faster at smaller creep strains, as shown in the following figure.
The predicted stress-strain response during uniaxial tension to a strain to 0.2 followed by unloading back to zero strain is shown in the following figure. The figure contains the results for 3 different m-values. The predicted stress response is similar in character to what is often seen for soft polymers.
The next figure shows the predicted tensile stress response of the Time Power model. The blue curve shows the stress response during a tension test to an engineering strain of 0.3. The red curve contains an initial hold of 1000 sec at zero strain, followed by the same tension test to an engineering strain of 0.3. This shows that a time-based creep model can give really strange results! In this case, holding the specimen in an unloaded configuration for 1000 seconds changes it response in later load segments. This is not what happens in real life.
Can Linear Elasticity with a Creep Model be used for Rubbers?
The following figure compares experimental cyclic tension data for a silicone rubber and the best predictions from a linear elastic model with a power creep model. The average error in the model predictions is 25.8%, and the predictions are not in good agreement with the experimental data. I do not recommend using this type of material model predicting the large strain response of rubbers. The TNV model from the PolyUMod library is a much better model in this case.
Can Linear Elasticity with a Creep Model be used for a Thermoplastic?
The following figure shows experimental cyclic data for ultrahigh molecular weight polyethylene (UHMWPE) and the best predictions from a linear elastic model with Power creep. The average error in the model predictions is 14.5%, but the predictions do not look particularly good. I do not recommend using this type of material model for predicting the large strain response of thermoplastics. The TNV model from the PolyUMod library is a much better model in this case.
Summary
- Do not use the “Time Law” or the “Strain Law” creep models in Abaqus.
- Try to avoid using a creep equation with explicit time dependence.
- Power Law creep model is my favorite.
- The creep models can be combined with linear elastic, and elastic-plastic components.
- The creep models are not great at predicting the response of polymers.
